91Ó°ÊÓ

Give a geometric description of the following sets of points. $$x^{2}+y^{2}+z^{2}-6 x+6 y-8 z-2=0$$

Let $$\mathbf{u}(t)=2 t^{3} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}-8 \mathbf{k} \text { and } \mathbf{v}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-e^{2 t} \mathbf{k}$$ Compute the derivative of the following functions. $$\left(4 t^{8}-6 t^{3}\right) \mathbf{v}(t)$$

Arc length of polar curves Find the length of the following polar curves. The complete cardioid \(r=2-2 \sin \theta\)

Let \(\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,\) and \(\mathbf{w}=\langle 0,8\rangle .\) Express the following vectors in the form \(\langle a, b\rangle\) $$10 \mathbf{u}-3 \mathbf{v}+\mathbf{w}$$

Find the cross products \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) for the following vectors \(\mathbf{u}\) and \(\mathbf{v}\) $$\mathbf{u}=\langle 3,-4,6\rangle, \mathbf{v}=\langle 1,2,-1\rangle$$

Graph the curves described by the following functions, indicating the direction of positive orientation. Try to anticipate the shape of the curve before using a graphing utility. $$\mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}+2 \mathbf{k}, \text { for } 0 \leq t \leq 2 \pi$$

Graph the curves described by the following functions, indicating the direction of positive orientation. Try to anticipate the shape of the curve before using a graphing utility. $$\mathbf{r}(t)=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}, \text { for } 0 \leq t \leq 6 \pi$$

Find the cross products \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) for the following vectors \(\mathbf{u}\) and \(\mathbf{v}\) $$\mathbf{u}=3 \mathbf{i}-\mathbf{j}-2 \mathbf{k}, \mathbf{v}=\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$$

Let \(\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,\) and \(\mathbf{w}=\langle 0,8\rangle .\) Express the following vectors in the form \(\langle a, b\rangle\) $$8 w+v-6 u$$

Give a geometric description of the following sets of points. $$x^{2}+y^{2}-14 y+z^{2} \geq-13$$